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  "body": "\n## Statement\n\nLet $\\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\\C$.\nFor every $z \\in \\C \\setminus \\set{0, -1, -2, \\ldots}$,\n$$\\Gamma(z) = \\lim_{n \\to \\infty} \\frac{n! \\; n^{z}}{z (z + 1) (z + 2) \\cdots (z + n)},$$\nwhere $n^z = e^{z \\ln n}$.\n\n## Remarks\n\nFor $\\operatorname{Re} z \u003e 0$, the quotient equals $\\int_0^n t^{z-1} (1 - t/n)^n \\, dt$, by repeated integration by parts.\nSince $(1 - t/n)^n$ increases to $e^{-t}$, the [[dominated convergence theorem|BYNV84PF86]] gives the limit $\\int_0^\\infty t^{z-1} e^{-t}\\,dt = \\Gamma(z)$.\nFor other $z$, the functional equation $\\Gamma(z + 1) = z\\Gamma(z)$ moves the argument into the half-plane, and the quotients transform in the same way up to a factor that tends to $1$.\nExample: at $z = 1$ the quotient is $\\frac{n!\\, n}{(n+1)!} = \\frac{n}{n+1} \\to 1 = \\Gamma(1)$.\nIn Mathlib the statement is `Complex.GammaSeq_tendsto_Gamma`.\n\n## Explanations\n\n### L1 human\n\nThe formula computes the gamma function as a limit of simple fractions built from factorials and powers.\nIt was Euler's first way to extend the factorial to non-integer arguments, and Gauss used it as the definition.\n\n### L2 human\n\nThe limit makes sense for every allowed complex $z$ at once, so it can serve as a definition of the meromorphic gamma function.\nTaking reciprocals and rewriting $n^z = \\exp\\big(z(\\ln n - 1 - \\tfrac12 - \\cdots - \\tfrac1n)\\big)\\exp\\big(z(1 + \\cdots + \\tfrac1n)\\big)$ turns it into the [[Weierstrass product|53MZ2DGDTQ]].\nIt is also the limit that appears in the proof of the [[Bohr–Mollerup theorem|QFF87AXWN2]].\n",
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